0.00/0.09	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.10	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.09/0.30	% Computer   : n015.cluster.edu
0.09/0.30	% Model      : x86_64 x86_64
0.09/0.30	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.09/0.30	% Memory     : 8042.1875MB
0.09/0.30	% OS         : Linux 3.10.0-693.el7.x86_64
0.09/0.30	% CPULimit   : 960
0.09/0.30	% WCLimit    : 120
0.09/0.30	% DateTime   : Thu Jul  2 06:52:15 EDT 2020
0.09/0.30	% CPUTime    : 
0.15/0.49	% SZS status Theorem
0.15/0.49	
0.15/0.49	% SZS output start Proof
0.15/0.49	Take the following subset of the input axioms:
0.15/0.49	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
0.15/0.49	  fof(axiom_m2, axiom, axiom_m2 <=> ![X, Y]: is_a_theorem(strict_implies(and(X, Y), X))).
0.15/0.49	  fof(hilbert_and_1, axiom, and_1).
0.15/0.49	  fof(km5_necessitation, axiom, necessitation).
0.15/0.49	  fof(necessitation, axiom, necessitation <=> ![X]: (is_a_theorem(X) => is_a_theorem(necessarily(X)))).
0.15/0.49	  fof(op_strict_implies, axiom, op_strict_implies => ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y)).
0.15/0.49	  fof(s1_0_axiom_m2, conjecture, axiom_m2).
0.15/0.49	  fof(s1_0_op_strict_implies, axiom, op_strict_implies).
0.15/0.49	
0.15/0.49	Now clausify the problem and encode Horn clauses using encoding 3 of
0.15/0.49	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.15/0.49	We repeatedly replace C & s=t => u=v by the two clauses:
0.15/0.49	  fresh(y, y, x1...xn) = u
0.15/0.49	  C => fresh(s, t, x1...xn) = v
0.15/0.49	where fresh is a fresh function symbol and x1..xn are the free
0.15/0.49	variables of u and v.
0.15/0.49	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.15/0.49	input problem has no model of domain size 1).
0.15/0.49	
0.15/0.49	The encoding turns the above axioms into the following unit equations and goals:
0.15/0.49	
0.15/0.49	Axiom 1 (and_1_1): fresh107(X, X, Y, Z) = true.
0.15/0.49	Axiom 2 (axiom_m2): fresh88(X, X) = true.
0.15/0.49	Axiom 3 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
0.15/0.49	Axiom 4 (necessitation_1): fresh33(X, X, Y) = true.
0.15/0.49	Axiom 5 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
0.15/0.49	Axiom 6 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
0.15/0.49	Axiom 7 (hilbert_and_1): and_1 = true.
0.15/0.49	Axiom 8 (axiom_m2_1): fresh87(axiom_m2, true, X, Y) = is_a_theorem(strict_implies(and(X, Y), X)).
0.15/0.49	Axiom 9 (axiom_m2): fresh88(is_a_theorem(strict_implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true) = axiom_m2.
0.15/0.49	Axiom 10 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
0.15/0.49	Axiom 11 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
0.15/0.49	Axiom 12 (km5_necessitation): necessitation = true.
0.15/0.50	Axiom 13 (s1_0_op_strict_implies): op_strict_implies = true.
0.15/0.50	
0.15/0.50	Goal 1 (s1_0_axiom_m2): axiom_m2 = true.
0.15/0.50	Proof:
0.15/0.50	  axiom_m2
0.15/0.50	= { by axiom 9 (axiom_m2) }
0.15/0.50	  fresh88(is_a_theorem(strict_implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 5 (op_strict_implies) }
0.15/0.50	  fresh88(is_a_theorem(fresh23(true, true, and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 13 (s1_0_op_strict_implies) }
0.15/0.50	  fresh88(is_a_theorem(fresh23(op_strict_implies, true, and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 11 (op_strict_implies) }
0.15/0.50	  fresh88(is_a_theorem(necessarily(implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X))), true)
0.15/0.50	= { by axiom 3 (necessitation_1) }
0.15/0.50	  fresh88(fresh34(true, true, implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 12 (km5_necessitation) }
0.15/0.50	  fresh88(fresh34(necessitation, true, implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 10 (necessitation_1) }
0.15/0.50	  fresh88(fresh33(is_a_theorem(implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true, implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 6 (and_1_1) }
0.15/0.50	  fresh88(fresh33(fresh107(and_1, true, sK22_axiom_m2_X, sK21_axiom_m2_Y), true, implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 7 (hilbert_and_1) }
0.15/0.50	  fresh88(fresh33(fresh107(true, true, sK22_axiom_m2_X, sK21_axiom_m2_Y), true, implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 1 (and_1_1) }
0.15/0.50	  fresh88(fresh33(true, true, implies(and(sK22_axiom_m2_X, sK21_axiom_m2_Y), sK22_axiom_m2_X)), true)
0.15/0.50	= { by axiom 4 (necessitation_1) }
0.15/0.50	  fresh88(true, true)
0.15/0.50	= { by axiom 2 (axiom_m2) }
0.15/0.50	  true
0.15/0.50	% SZS output end Proof
0.15/0.50	
0.15/0.50	RESULT: Theorem (the conjecture is true).
0.15/0.50	EOF